In reply to 1234None:
Do you have to remain based in KK or could you travel on somewhere and stay somewhere a short flight/drive away? If so how long are you in the area for and what else are you doing whilst your in Borneo then I can probably suggest a few options to look at. I live in Miri, Sarawak (on Borneo, just the other side of Brunei).
With regards to the comments on guides and similar. If you're coming from outside the country it's much easier to just book organised tours with a guide. You can just turn up to lots of these places without a guide but be aware information published on websites (feeding times for Sepilok, opening hours, etc) is often wrong. Sepilok and the Proboscis monkey sanctuary you could just turn up without a guide. Phone them first to check the feeding times though.
Mount Kinabalu you would need a guide for.
http://www.mountaintorq.com/location came up with a good package for me including the summit, two days of guided climbing (bolt clipping) and the via feratta. Haven't actually had the time to go yet but they were really helpful. I would also ask to do the longer walk in (same altitude gain but longer distance) as it's supposed to be a much nicer walk.
Climbing options wise have a chat with the guys at
http://sabahbah.com/activities/sabah-indoor-climbing-centre/. They'll be able to advise on where you can go outside. I would plan to climb either early morning or around dusk (though wear lots of bug spray) or at night due to the temperatures. Trying to climb in the midday heat is just not worthwhile (especially if your trying something small and crimpy.
Mike